Q. 109 - Precalculus Enhanced with Graphing Utilities

Question

To approximate the speed of the current of a river, a circular paddle wheel with radius 4 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 10 revolutions per minute, what is the speed of the current? Express your answer in miles per hour. 

Step-by-Step Solution

Answer

 The speed of the current is 2.85 miles per hour. 

Step 1: Given information

The current of the river causes the wheel to rotate at a speed of 10 revolutions per minute.

The radius of the circular paddle is 4 feet.

Step 2: Calculate the speed of the current.

Convert angular speed into linear speed.

$$\begin{gathered} 1 \mathrm{revolution} = 2\mathrm{\pi r} \\ 10 \mathrm{revolution}=10\times 2\mathrm{\pi r} \\ 10 \mathrm{revolution}=20\mathrm{\pi r} \\ 10 \mathrm{revolution}=20\times 3.14\times 4=251.2 \mathrm{feet} \end{gathered}$$

So that,

$$\begin{gathered} 10 \mathrm{rev}. \mathrm{per} \mathrm{minute}=\frac{251.2 \mathrm{feet}}{{\displaystyle \frac{1}{60}}\mathrm{hour}}=15072\frac{\mathrm{feet}}{\mathrm{hour}} \\ =\frac{15072}{5280}\frac{\mathrm{miles}}{\mathrm{hour}} \\ =2.85 \frac{\mathrm{miles}}{\mathrm{hour}} \end{gathered}$$

Thus, the speed of the current is 2.85 miles per hour.